LMFDB - Embedded newform 810.4.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(1,810)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("810.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8657424.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 125x^{2} + 126x + 3726 $ x^4 - 2x^3 - 125x^2 + 126*x + 3726
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.40373$ of defining polynomial
Character	$\chi$	$=$	810.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -9.81890 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -9.81890 q^{7} +8.00000 q^{8} -10.0000 q^{10} +19.0786 q^{11} -87.4818 q^{13} -19.6378 q^{14} +16.0000 q^{16} -10.8482 q^{17} +103.354 q^{19} -20.0000 q^{20} +38.1573 q^{22} +196.194 q^{23} +25.0000 q^{25} -174.964 q^{26} -39.2756 q^{28} +12.0983 q^{29} +234.840 q^{31} +32.0000 q^{32} -21.6964 q^{34} +49.0945 q^{35} +304.576 q^{37} +206.708 q^{38} -40.0000 q^{40} -364.759 q^{41} +339.517 q^{43} +76.3145 q^{44} +392.388 q^{46} +237.129 q^{47} -246.589 q^{49} +50.0000 q^{50} -349.927 q^{52} +31.2915 q^{53} -95.3931 q^{55} -78.5512 q^{56} +24.1966 q^{58} -129.694 q^{59} -538.940 q^{61} +469.680 q^{62} +64.0000 q^{64} +437.409 q^{65} +337.061 q^{67} -43.3927 q^{68} +98.1890 q^{70} +264.767 q^{71} +950.225 q^{73} +609.151 q^{74} +413.415 q^{76} -187.331 q^{77} +375.097 q^{79} -80.0000 q^{80} -729.519 q^{82} +1077.10 q^{83} +54.2409 q^{85} +679.035 q^{86} +152.629 q^{88} -909.671 q^{89} +858.975 q^{91} +784.776 q^{92} +474.258 q^{94} -516.769 q^{95} +1317.71 q^{97} -493.179 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8} - 40 q^{10} - 36 q^{11} + 32 q^{13} + 4 q^{14} + 64 q^{16} + 90 q^{17} + 164 q^{19} - 80 q^{20} - 72 q^{22} + 42 q^{23} + 100 q^{25} + 64 q^{26} + 8 q^{28} + 36 q^{29} + 446 q^{31} + 128 q^{32} + 180 q^{34} - 10 q^{35} + 686 q^{37} + 328 q^{38} - 160 q^{40} + 108 q^{41} + 470 q^{43} - 144 q^{44} + 84 q^{46} + 804 q^{47} + 1338 q^{49} + 200 q^{50} + 128 q^{52} + 168 q^{53} + 180 q^{55} + 16 q^{56} + 72 q^{58} + 768 q^{59} + 596 q^{61} + 892 q^{62} + 256 q^{64} - 160 q^{65} - 424 q^{67} + 360 q^{68} - 20 q^{70} + 348 q^{71} - 94 q^{73} + 1372 q^{74} + 656 q^{76} + 630 q^{77} + 1088 q^{79} - 320 q^{80} + 216 q^{82} + 522 q^{83} - 450 q^{85} + 940 q^{86} - 288 q^{88} + 1128 q^{89} + 2842 q^{91} + 168 q^{92} + 1608 q^{94} - 820 q^{95} + 2486 q^{97} + 2676 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8 - 40 * q^10 - 36 * q^11 + 32 * q^13 + 4 * q^14 + 64 * q^16 + 90 * q^17 + 164 * q^19 - 80 * q^20 - 72 * q^22 + 42 * q^23 + 100 * q^25 + 64 * q^26 + 8 * q^28 + 36 * q^29 + 446 * q^31 + 128 * q^32 + 180 * q^34 - 10 * q^35 + 686 * q^37 + 328 * q^38 - 160 * q^40 + 108 * q^41 + 470 * q^43 - 144 * q^44 + 84 * q^46 + 804 * q^47 + 1338 * q^49 + 200 * q^50 + 128 * q^52 + 168 * q^53 + 180 * q^55 + 16 * q^56 + 72 * q^58 + 768 * q^59 + 596 * q^61 + 892 * q^62 + 256 * q^64 - 160 * q^65 - 424 * q^67 + 360 * q^68 - 20 * q^70 + 348 * q^71 - 94 * q^73 + 1372 * q^74 + 656 * q^76 + 630 * q^77 + 1088 * q^79 - 320 * q^80 + 216 * q^82 + 522 * q^83 - 450 * q^85 + 940 * q^86 - 288 * q^88 + 1128 * q^89 + 2842 * q^91 + 168 * q^92 + 1608 * q^94 - 820 * q^95 + 2486 * q^97 + 2676 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−9.81890	−0.530171	−0.265085	−	0.964225i	$-0.585400\pi$
$7$	−9.81890	−0.530171	−0.265085	+	0.964225i	$0.585400\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	19.0786	0.522947	0.261474	−	0.965211i	$-0.415792\pi$
$11$	19.0786	0.522947	0.261474	+	0.965211i	$0.415792\pi$
$12$	0	0
$13$	−87.4818	−1.86639	−0.933196	−	0.359369i	$-0.882992\pi$
$13$	−87.4818	−1.86639	−0.933196	+	0.359369i	$0.882992\pi$
$14$	−19.6378	−0.374887
$15$	0	0
$16$	16.0000	0.250000
$17$	−10.8482	−0.154769	−0.0773845	−	0.997001i	$-0.524657\pi$
$17$	−10.8482	−0.154769	−0.0773845	+	0.997001i	$0.524657\pi$
$18$	0	0
$19$	103.354	1.24795	0.623974	−	0.781445i	$-0.285517\pi$
$19$	103.354	1.24795	0.623974	+	0.781445i	$0.285517\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	38.1573	0.369780
$23$	196.194	1.77866	0.889332	−	0.457262i	$-0.151170\pi$
$23$	196.194	1.77866	0.889332	+	0.457262i	$0.151170\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−174.964	−1.31974
$27$	0	0
$28$	−39.2756	−0.265085
$29$	12.0983	0.0774688	0.0387344	−	0.999250i	$-0.487667\pi$
$29$	12.0983	0.0774688	0.0387344	+	0.999250i	$0.487667\pi$
$30$	0	0
$31$	234.840	1.36060	0.680299	−	0.732935i	$-0.261850\pi$
$31$	234.840	1.36060	0.680299	+	0.732935i	$0.261850\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−21.6964	−0.109438
$35$	49.0945	0.237100
$36$	0	0
$37$	304.576	1.35329	0.676647	−	0.736307i	$-0.263432\pi$
$37$	304.576	1.35329	0.676647	+	0.736307i	$0.263432\pi$
$38$	206.708	0.882432
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−364.759	−1.38941	−0.694706	−	0.719294i	$-0.744465\pi$
$41$	−364.759	−1.38941	−0.694706	+	0.719294i	$0.744465\pi$
$42$	0	0
$43$	339.517	1.20409	0.602045	−	0.798462i	$-0.294353\pi$
$43$	339.517	1.20409	0.602045	+	0.798462i	$0.294353\pi$
$44$	76.3145	0.261474
$45$	0	0
$46$	392.388	1.25771
$47$	237.129	0.735933	0.367967	−	0.929839i	$-0.380054\pi$
$47$	237.129	0.735933	0.367967	+	0.929839i	$0.380054\pi$
$48$	0	0
$49$	−246.589	−0.718919
$50$	50.0000	0.141421
$51$	0	0
$52$	−349.927	−0.933196
$53$	31.2915	0.0810986	0.0405493	−	0.999178i	$-0.487089\pi$
$53$	31.2915	0.0810986	0.0405493	+	0.999178i	$0.487089\pi$
$54$	0	0
$55$	−95.3931	−0.233869
$56$	−78.5512	−0.187444
$57$	0	0
$58$	24.1966	0.0547787
$59$	−129.694	−0.286181	−0.143091	−	0.989710i	$-0.545704\pi$
$59$	−129.694	−0.286181	−0.143091	+	0.989710i	$0.545704\pi$
$60$	0	0
$61$	−538.940	−1.13122	−0.565608	−	0.824674i	$-0.691359\pi$
$61$	−538.940	−1.13122	−0.565608	+	0.824674i	$0.691359\pi$
$62$	469.680	0.962088
$63$	0	0
$64$	64.0000	0.125000
$65$	437.409	0.834675
$66$	0	0
$67$	337.061	0.614606	0.307303	−	0.951612i	$-0.400573\pi$
$67$	337.061	0.614606	0.307303	+	0.951612i	$0.400573\pi$
$68$	−43.3927	−0.0773845
$69$	0	0
$70$	98.1890	0.167655
$71$	264.767	0.442564	0.221282	−	0.975210i	$-0.428976\pi$
$71$	264.767	0.442564	0.221282	+	0.975210i	$0.428976\pi$
$72$	0	0
$73$	950.225	1.52350	0.761749	−	0.647872i	$-0.224341\pi$
$73$	950.225	1.52350	0.761749	+	0.647872i	$0.224341\pi$
$74$	609.151	0.956924
$75$	0	0
$76$	413.415	0.623974
$77$	−187.331	−0.277251
$78$	0	0
$79$	375.097	0.534198	0.267099	−	0.963669i	$-0.413935\pi$
$79$	375.097	0.534198	0.267099	+	0.963669i	$0.413935\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	−729.519	−0.982462
$83$	1077.10	1.42443	0.712214	−	0.701962i	$-0.247692\pi$
$83$	1077.10	1.42443	0.712214	+	0.701962i	$0.247692\pi$
$84$	0	0
$85$	54.2409	0.0692148
$86$	679.035	0.851421
$87$	0	0
$88$	152.629	0.184890
$89$	−909.671	−1.08343	−0.541713	−	0.840564i	$-0.682224\pi$
$89$	−909.671	−1.08343	−0.541713	+	0.840564i	$0.682224\pi$
$90$	0	0
$91$	858.975	0.989506
$92$	784.776	0.889332
$93$	0	0
$94$	474.258	0.520383
$95$	−516.769	−0.558099
$96$	0	0
$97$	1317.71	1.37931	0.689654	−	0.724139i	$-0.257763\pi$
$97$	1317.71	1.37931	0.689654	+	0.724139i	$0.257763\pi$
$98$	−493.179	−0.508353
$99$	0	0
$100$	100.000	0.100000
$101$	642.982	0.633456	0.316728	−	0.948516i	$-0.397416\pi$
$101$	642.982	0.633456	0.316728	+	0.948516i	$0.397416\pi$
$102$	0	0
$103$	55.3655	0.0529643	0.0264821	−	0.999649i	$-0.491569\pi$
$103$	55.3655	0.0529643	0.0264821	+	0.999649i	$0.491569\pi$
$104$	−699.854	−0.659869
$105$	0	0
$106$	62.5831	0.0573453
$107$	−178.586	−0.161351	−0.0806757	−	0.996740i	$-0.525708\pi$
$107$	−178.586	−0.161351	−0.0806757	+	0.996740i	$0.525708\pi$
$108$	0	0
$109$	723.780	0.636014	0.318007	−	0.948088i	$-0.396986\pi$
$109$	723.780	0.636014	0.318007	+	0.948088i	$0.396986\pi$
$110$	−190.786	−0.165371
$111$	0	0
$112$	−157.102	−0.132543
$113$	−1182.26	−0.984229	−0.492114	−	0.870531i	$-0.663776\pi$
$113$	−1182.26	−0.984229	−0.492114	+	0.870531i	$0.663776\pi$
$114$	0	0
$115$	−980.970	−0.795443
$116$	48.3932	0.0387344
$117$	0	0
$118$	−259.387	−0.202361
$119$	106.517	0.0820539
$120$	0	0
$121$	−967.006	−0.726526
$122$	−1077.88	−0.799891
$123$	0	0
$124$	939.361	0.680299
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−232.957	−0.162769	−0.0813844	−	0.996683i	$-0.525934\pi$
$127$	−232.957	−0.162769	−0.0813844	+	0.996683i	$0.525934\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	874.818	0.590205
$131$	−2150.85	−1.43451	−0.717254	−	0.696812i	$-0.754601\pi$
$131$	−2150.85	−1.43451	−0.717254	+	0.696812i	$0.754601\pi$
$132$	0	0
$133$	−1014.82	−0.661625
$134$	674.123	0.434592
$135$	0	0
$136$	−86.7855	−0.0547191
$137$	2351.74	1.46659	0.733294	−	0.679912i	$-0.237982\pi$
$137$	2351.74	1.46659	0.733294	+	0.679912i	$0.237982\pi$
$138$	0	0
$139$	1069.20	0.652435	0.326218	−	0.945295i	$-0.394226\pi$
$139$	1069.20	0.652435	0.326218	+	0.945295i	$0.394226\pi$
$140$	196.378	0.118550
$141$	0	0
$142$	529.533	0.312940
$143$	−1669.03	−0.976024
$144$	0	0
$145$	−60.4914	−0.0346451
$146$	1900.45	1.07728
$147$	0	0
$148$	1218.30	0.676647
$149$	311.846	0.171459	0.0857295	−	0.996318i	$-0.472678\pi$
$149$	311.846	0.171459	0.0857295	+	0.996318i	$0.472678\pi$
$150$	0	0
$151$	2534.43	1.36589	0.682943	−	0.730471i	$-0.260700\pi$
$151$	2534.43	1.36589	0.682943	+	0.730471i	$0.260700\pi$
$152$	826.831	0.441216
$153$	0	0
$154$	−374.662	−0.196046
$155$	−1174.20	−0.608478
$156$	0	0
$157$	−1057.34	−0.537484	−0.268742	−	0.963212i	$-0.586608\pi$
$157$	−1057.34	−0.537484	−0.268742	+	0.963212i	$0.586608\pi$
$158$	750.193	0.377735
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−1926.41	−0.942996
$162$	0	0
$163$	2665.14	1.28067	0.640337	−	0.768094i	$-0.278795\pi$
$163$	2665.14	1.28067	0.640337	+	0.768094i	$0.278795\pi$
$164$	−1459.04	−0.694706
$165$	0	0
$166$	2154.21	1.00722
$167$	−1534.51	−0.711040	−0.355520	−	0.934669i	$-0.615696\pi$
$167$	−1534.51	−0.711040	−0.355520	+	0.934669i	$0.615696\pi$
$168$	0	0
$169$	5456.06	2.48342
$170$	108.482	0.0489422
$171$	0	0
$172$	1358.07	0.602045
$173$	−2007.84	−0.882391	−0.441195	−	0.897411i	$-0.645445\pi$
$173$	−2007.84	−0.882391	−0.441195	+	0.897411i	$0.645445\pi$
$174$	0	0
$175$	−245.472	−0.106034
$176$	305.258	0.130737
$177$	0	0
$178$	−1819.34	−0.766098
$179$	−4588.33	−1.91591	−0.957955	−	0.286917i	$-0.907370\pi$
$179$	−4588.33	−1.91591	−0.957955	+	0.286917i	$0.907370\pi$
$180$	0	0
$181$	15.7678	0.00647519	0.00323759	−	0.999995i	$-0.498969\pi$
$181$	15.7678	0.00647519	0.00323759	+	0.999995i	$0.498969\pi$
$182$	1717.95	0.699686
$183$	0	0
$184$	1569.55	0.628853
$185$	−1522.88	−0.605212
$186$	0	0
$187$	−206.968	−0.0809360
$188$	948.517	0.367967
$189$	0	0
$190$	−1033.54	−0.394636
$191$	−1597.73	−0.605275	−0.302637	−	0.953106i	$-0.597867\pi$
$191$	−1597.73	−0.605275	−0.302637	+	0.953106i	$0.597867\pi$
$192$	0	0
$193$	579.868	0.216268	0.108134	−	0.994136i	$-0.465512\pi$
$193$	579.868	0.216268	0.108134	+	0.994136i	$0.465512\pi$
$194$	2635.41	0.975317
$195$	0	0
$196$	−986.357	−0.359460
$197$	2095.96	0.758026	0.379013	−	0.925391i	$-0.376264\pi$
$197$	2095.96	0.758026	0.379013	+	0.925391i	$0.376264\pi$
$198$	0	0
$199$	−1748.58	−0.622881	−0.311440	−	0.950266i	$-0.600811\pi$
$199$	−1748.58	−0.622881	−0.311440	+	0.950266i	$0.600811\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	1285.96	0.447921
$203$	−118.792	−0.0410717
$204$	0	0
$205$	1823.80	0.621364
$206$	110.731	0.0374514
$207$	0	0
$208$	−1399.71	−0.466598
$209$	1971.85	0.652611
$210$	0	0
$211$	1122.66	0.366291	0.183145	−	0.983086i	$-0.441372\pi$
$211$	1122.66	0.366291	0.183145	+	0.983086i	$0.441372\pi$
$212$	125.166	0.0405493
$213$	0	0
$214$	−357.173	−0.114093
$215$	−1697.59	−0.538486
$216$	0	0
$217$	−2305.87	−0.721349
$218$	1447.56	0.449730
$219$	0	0
$220$	−381.573	−0.116935
$221$	949.019	0.288859
$222$	0	0
$223$	4452.63	1.33709	0.668543	−	0.743673i	$-0.266918\pi$
$223$	4452.63	1.33709	0.668543	+	0.743673i	$0.266918\pi$
$224$	−314.205	−0.0937218
$225$	0	0
$226$	−2364.52	−0.695955
$227$	−3271.60	−0.956581	−0.478290	−	0.878202i	$-0.658743\pi$
$227$	−3271.60	−0.956581	−0.478290	+	0.878202i	$0.658743\pi$
$228$	0	0
$229$	77.3289	0.0223146	0.0111573	−	0.999938i	$-0.496448\pi$
$229$	77.3289	0.0223146	0.0111573	+	0.999938i	$0.496448\pi$
$230$	−1961.94	−0.562463
$231$	0	0
$232$	96.7863	0.0273894
$233$	4243.59	1.19316	0.596581	−	0.802553i	$-0.296525\pi$
$233$	4243.59	1.19316	0.596581	+	0.802553i	$0.296525\pi$
$234$	0	0
$235$	−1185.65	−0.329119
$236$	−518.775	−0.143091
$237$	0	0
$238$	213.034	0.0580209
$239$	−78.4946	−0.0212443	−0.0106222	−	0.999944i	$-0.503381\pi$
$239$	−78.4946	−0.0212443	−0.0106222	+	0.999944i	$0.503381\pi$
$240$	0	0
$241$	−5397.70	−1.44272	−0.721361	−	0.692559i	$-0.756483\pi$
$241$	−5397.70	−1.44272	−0.721361	+	0.692559i	$0.756483\pi$
$242$	−1934.01	−0.513731
$243$	0	0
$244$	−2155.76	−0.565608
$245$	1232.95	0.321510
$246$	0	0
$247$	−9041.58	−2.32916
$248$	1878.72	0.481044
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−1627.51	−0.409274	−0.204637	−	0.978838i	$-0.565601\pi$
$251$	−1627.51	−0.409274	−0.204637	+	0.978838i	$0.565601\pi$
$252$	0	0
$253$	3743.11	0.930148
$254$	−465.915	−0.115095
$255$	0	0
$256$	256.000	0.0625000
$257$	−6183.15	−1.50076	−0.750378	−	0.661009i	$-0.770128\pi$
$257$	−6183.15	−1.50076	−0.750378	+	0.661009i	$0.770128\pi$
$258$	0	0
$259$	−2990.60	−0.717477
$260$	1749.64	0.417338
$261$	0	0
$262$	−4301.70	−1.01435
$263$	7289.27	1.70903	0.854517	−	0.519424i	$-0.173853\pi$
$263$	7289.27	1.70903	0.854517	+	0.519424i	$0.173853\pi$
$264$	0	0
$265$	−156.458	−0.0362684
$266$	−2029.64	−0.467839
$267$	0	0
$268$	1348.25	0.307303
$269$	−1903.32	−0.431404	−0.215702	−	0.976459i	$-0.569204\pi$
$269$	−1903.32	−0.431404	−0.215702	+	0.976459i	$0.569204\pi$
$270$	0	0
$271$	6485.88	1.45383	0.726917	−	0.686725i	$-0.240952\pi$
$271$	6485.88	1.45383	0.726917	+	0.686725i	$0.240952\pi$
$272$	−173.571	−0.0386922
$273$	0	0
$274$	4703.48	1.03703
$275$	476.966	0.104589
$276$	0	0
$277$	−5847.78	−1.26844	−0.634222	−	0.773151i	$-0.718679\pi$
$277$	−5847.78	−1.26844	−0.634222	+	0.773151i	$0.718679\pi$
$278$	2138.40	0.461341
$279$	0	0
$280$	392.756	0.0838273
$281$	−3736.13	−0.793163	−0.396581	−	0.918000i	$-0.629804\pi$
$281$	−3736.13	−0.793163	−0.396581	+	0.918000i	$0.629804\pi$
$282$	0	0
$283$	−2860.11	−0.600762	−0.300381	−	0.953819i	$-0.597114\pi$
$283$	−2860.11	−0.600762	−0.300381	+	0.953819i	$0.597114\pi$
$284$	1059.07	0.221282
$285$	0	0
$286$	−3338.06	−0.690153
$287$	3581.54	0.736625
$288$	0	0
$289$	−4795.32	−0.976047
$290$	−120.983	−0.0244978
$291$	0	0
$292$	3800.90	0.761749
$293$	8038.74	1.60283	0.801413	−	0.598111i	$-0.204082\pi$
$293$	8038.74	1.60283	0.801413	+	0.598111i	$0.204082\pi$
$294$	0	0
$295$	648.469	0.127984
$296$	2436.60	0.478462
$297$	0	0
$298$	623.691	0.121240
$299$	−17163.4	−3.31968
$300$	0	0
$301$	−3333.69	−0.638374
$302$	5068.86	0.965828
$303$	0	0
$304$	1653.66	0.311987
$305$	2694.70	0.505895
$306$	0	0
$307$	3066.57	0.570093	0.285046	−	0.958514i	$-0.407991\pi$
$307$	3066.57	0.570093	0.285046	+	0.958514i	$0.407991\pi$
$308$	−749.324	−0.138626
$309$	0	0
$310$	−2348.40	−0.430259
$311$	9729.22	1.77393	0.886967	−	0.461834i	$-0.152808\pi$
$311$	9729.22	1.77393	0.886967	+	0.461834i	$0.152808\pi$
$312$	0	0
$313$	−717.127	−0.129503	−0.0647515	−	0.997901i	$-0.520625\pi$
$313$	−717.127	−0.129503	−0.0647515	+	0.997901i	$0.520625\pi$
$314$	−2114.68	−0.380058
$315$	0	0
$316$	1500.39	0.267099
$317$	−3132.05	−0.554932	−0.277466	−	0.960735i	$-0.589495\pi$
$317$	−3132.05	−0.554932	−0.277466	+	0.960735i	$0.589495\pi$
$318$	0	0
$319$	230.819	0.0405121
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−3852.82	−0.666799
$323$	−1121.20	−0.193143
$324$	0	0
$325$	−2187.04	−0.373278
$326$	5330.28	0.905574
$327$	0	0
$328$	−2918.08	−0.491231
$329$	−2328.35	−0.390170
$330$	0	0
$331$	−4822.14	−0.800752	−0.400376	−	0.916351i	$-0.631120\pi$
$331$	−4822.14	−0.800752	−0.400376	+	0.916351i	$0.631120\pi$
$332$	4308.42	0.712214
$333$	0	0
$334$	−3069.01	−0.502781
$335$	−1685.31	−0.274860
$336$	0	0
$337$	−6360.78	−1.02817	−0.514086	−	0.857739i	$-0.671869\pi$
$337$	−6360.78	−1.02817	−0.514086	+	0.857739i	$0.671869\pi$
$338$	10912.1	1.75604
$339$	0	0
$340$	216.964	0.0346074
$341$	4480.43	0.711521
$342$	0	0
$343$	5789.12	0.911320
$344$	2716.14	0.425710
$345$	0	0
$346$	−4015.69	−0.623944
$347$	−10330.7	−1.59821	−0.799104	−	0.601192i	$-0.794693\pi$
$347$	−10330.7	−1.59821	−0.799104	+	0.601192i	$0.794693\pi$
$348$	0	0
$349$	9630.71	1.47713	0.738567	−	0.674180i	$-0.235503\pi$
$349$	9630.71	1.47713	0.738567	+	0.674180i	$0.235503\pi$
$350$	−490.945	−0.0749774
$351$	0	0
$352$	610.516	0.0924449
$353$	−11395.2	−1.71814	−0.859070	−	0.511858i	$-0.828957\pi$
$353$	−11395.2	−1.71814	−0.859070	+	0.511858i	$0.828957\pi$
$354$	0	0
$355$	−1323.83	−0.197920
$356$	−3638.68	−0.541713
$357$	0	0
$358$	−9176.66	−1.35475
$359$	1549.64	0.227819	0.113909	−	0.993491i	$-0.463663\pi$
$359$	1549.64	0.227819	0.113909	+	0.993491i	$0.463663\pi$
$360$	0	0
$361$	3823.01	0.557372
$362$	31.5355	0.00457865
$363$	0	0
$364$	3435.90	0.494753
$365$	−4751.12	−0.681329
$366$	0	0
$367$	−5046.78	−0.717820	−0.358910	−	0.933372i	$-0.616851\pi$
$367$	−5046.78	−0.717820	−0.358910	+	0.933372i	$0.616851\pi$
$368$	3139.10	0.444666
$369$	0	0
$370$	−3045.76	−0.427949
$371$	−307.248	−0.0429961
$372$	0	0
$373$	−10184.0	−1.41370	−0.706849	−	0.707365i	$-0.749884\pi$
$373$	−10184.0	−1.41370	−0.706849	+	0.707365i	$0.749884\pi$
$374$	−413.937	−0.0572304
$375$	0	0
$376$	1897.03	0.260192
$377$	−1058.38	−0.144587
$378$	0	0
$379$	−7332.90	−0.993841	−0.496921	−	0.867796i	$-0.665536\pi$
$379$	−7332.90	−0.993841	−0.496921	+	0.867796i	$0.665536\pi$
$380$	−2067.08	−0.279049
$381$	0	0
$382$	−3195.45	−0.427994
$383$	1085.50	0.144821	0.0724103	−	0.997375i	$-0.476931\pi$
$383$	1085.50	0.144821	0.0724103	+	0.997375i	$0.476931\pi$
$384$	0	0
$385$	936.655	0.123991
$386$	1159.74	0.152925
$387$	0	0
$388$	5270.83	0.689654
$389$	427.888	0.0557707	0.0278853	−	0.999611i	$-0.491123\pi$
$389$	427.888	0.0557707	0.0278853	+	0.999611i	$0.491123\pi$
$390$	0	0
$391$	−2128.35	−0.275282
$392$	−1972.71	−0.254176
$393$	0	0
$394$	4191.92	0.536005
$395$	−1875.48	−0.238901
$396$	0	0
$397$	−2668.85	−0.337394	−0.168697	−	0.985668i	$-0.553956\pi$
$397$	−2668.85	−0.337394	−0.168697	+	0.985668i	$0.553956\pi$
$398$	−3497.15	−0.440443
$399$	0	0
$400$	400.000	0.0500000
$401$	11935.2	1.48632	0.743160	−	0.669113i	$-0.233326\pi$
$401$	11935.2	1.48632	0.743160	+	0.669113i	$0.233326\pi$
$402$	0	0
$403$	−20544.2	−2.53941
$404$	2571.93	0.316728
$405$	0	0
$406$	−237.584	−0.0290421
$407$	5810.88	0.707702
$408$	0	0
$409$	−766.424	−0.0926582	−0.0463291	−	0.998926i	$-0.514752\pi$
$409$	−766.424	−0.0926582	−0.0463291	+	0.998926i	$0.514752\pi$
$410$	3647.59	0.439370
$411$	0	0
$412$	221.462	0.0264821
$413$	1273.45	0.151725
$414$	0	0
$415$	−5385.52	−0.637024
$416$	−2799.42	−0.329934
$417$	0	0
$418$	3943.70	0.461465
$419$	11634.1	1.35647	0.678237	−	0.734844i	$-0.262744\pi$
$419$	11634.1	1.35647	0.678237	+	0.734844i	$0.262744\pi$
$420$	0	0
$421$	11638.8	1.34737	0.673683	−	0.739020i	$-0.264711\pi$
$421$	11638.8	1.34737	0.673683	+	0.739020i	$0.264711\pi$
$422$	2245.33	0.259007
$423$	0	0
$424$	250.332	0.0286727
$425$	−271.205	−0.0309538
$426$	0	0
$427$	5291.80	0.599738
$428$	−714.346	−0.0806757
$429$	0	0
$430$	−3395.17	−0.380767
$431$	7408.63	0.827984	0.413992	−	0.910280i	$-0.364134\pi$
$431$	7408.63	0.827984	0.413992	+	0.910280i	$0.364134\pi$
$432$	0	0
$433$	9830.81	1.09108	0.545541	−	0.838084i	$-0.316324\pi$
$433$	9830.81	1.09108	0.545541	+	0.838084i	$0.316324\pi$
$434$	−4611.74	−0.510071
$435$	0	0
$436$	2895.12	0.318007
$437$	20277.4	2.21968
$438$	0	0
$439$	−3731.68	−0.405703	−0.202851	−	0.979210i	$-0.565021\pi$
$439$	−3731.68	−0.405703	−0.202851	+	0.979210i	$0.565021\pi$
$440$	−763.145	−0.0826853
$441$	0	0
$442$	1898.04	0.204254
$443$	8134.88	0.872460	0.436230	−	0.899835i	$-0.356313\pi$
$443$	8134.88	0.872460	0.436230	+	0.899835i	$0.356313\pi$
$444$	0	0
$445$	4548.35	0.484523
$446$	8905.26	0.945463
$447$	0	0
$448$	−628.409	−0.0662713
$449$	−17834.3	−1.87450	−0.937250	−	0.348658i	$-0.886637\pi$
$449$	−17834.3	−1.87450	−0.937250	+	0.348658i	$0.886637\pi$
$450$	0	0
$451$	−6959.11	−0.726589
$452$	−4729.05	−0.492114
$453$	0	0
$454$	−6543.20	−0.676405
$455$	−4294.87	−0.442520
$456$	0	0
$457$	−4956.69	−0.507362	−0.253681	−	0.967288i	$-0.581641\pi$
$457$	−4956.69	−0.507362	−0.253681	+	0.967288i	$0.581641\pi$
$458$	154.658	0.0157788
$459$	0	0
$460$	−3923.88	−0.397721
$461$	−3588.53	−0.362548	−0.181274	−	0.983433i	$-0.558022\pi$
$461$	−3588.53	−0.362548	−0.181274	+	0.983433i	$0.558022\pi$
$462$	0	0
$463$	10501.0	1.05405	0.527024	−	0.849850i	$-0.323308\pi$
$463$	10501.0	1.05405	0.527024	+	0.849850i	$0.323308\pi$
$464$	193.573	0.0193672
$465$	0	0
$466$	8487.18	0.843693
$467$	11303.2	1.12002	0.560009	−	0.828486i	$-0.310798\pi$
$467$	11303.2	1.12002	0.560009	+	0.828486i	$0.310798\pi$
$468$	0	0
$469$	−3309.57	−0.325846
$470$	−2371.29	−0.232722
$471$	0	0
$472$	−1037.55	−0.101180
$473$	6477.52	0.629676
$474$	0	0
$475$	2583.85	0.249589
$476$	426.069	0.0410270
$477$	0	0
$478$	−156.989	−0.0150220
$479$	−17292.5	−1.64951	−0.824753	−	0.565493i	$-0.808686\pi$
$479$	−17292.5	−1.64951	−0.824753	+	0.565493i	$0.808686\pi$
$480$	0	0
$481$	−26644.8	−2.52578
$482$	−10795.4	−1.02016
$483$	0	0
$484$	−3868.02	−0.363263
$485$	−6588.53	−0.616845
$486$	0	0
$487$	1617.93	0.150545	0.0752724	−	0.997163i	$-0.476017\pi$
$487$	1617.93	0.150545	0.0752724	+	0.997163i	$0.476017\pi$
$488$	−4311.52	−0.399946
$489$	0	0
$490$	2465.89	0.227342
$491$	−2741.69	−0.251998	−0.125999	−	0.992030i	$-0.540214\pi$
$491$	−2741.69	−0.251998	−0.125999	+	0.992030i	$0.540214\pi$
$492$	0	0
$493$	−131.244	−0.0119898
$494$	−18083.2	−1.64696
$495$	0	0
$496$	3757.44	0.340150
$497$	−2599.72	−0.234634
$498$	0	0
$499$	13757.2	1.23418	0.617090	−	0.786893i	$-0.288311\pi$
$499$	13757.2	1.23418	0.617090	+	0.786893i	$0.288311\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	−3255.03	−0.289400
$503$	−5929.32	−0.525597	−0.262798	−	0.964851i	$-0.584645\pi$
$503$	−5929.32	−0.525597	−0.262798	+	0.964851i	$0.584645\pi$
$504$	0	0
$505$	−3214.91	−0.283290
$506$	7486.22	0.657714
$507$	0	0
$508$	−931.830	−0.0813844
$509$	−16551.6	−1.44133	−0.720666	−	0.693282i	$-0.756164\pi$
$509$	−16551.6	−1.44133	−0.720666	+	0.693282i	$0.756164\pi$
$510$	0	0
$511$	−9330.16	−0.807714
$512$	512.000	0.0441942
$513$	0	0
$514$	−12366.3	−1.06119
$515$	−276.827	−0.0236863
$516$	0	0
$517$	4524.10	0.384854
$518$	−5981.19	−0.507333
$519$	0	0
$520$	3499.27	0.295102
$521$	20294.3	1.70654	0.853271	−	0.521468i	$-0.174616\pi$
$521$	20294.3	1.70654	0.853271	+	0.521468i	$0.174616\pi$
$522$	0	0
$523$	21068.6	1.76150	0.880752	−	0.473578i	$-0.157038\pi$
$523$	21068.6	1.76150	0.880752	+	0.473578i	$0.157038\pi$
$524$	−8603.39	−0.717254
$525$	0	0
$526$	14578.5	1.20847
$527$	−2547.59	−0.210578
$528$	0	0
$529$	26325.1	2.16365
$530$	−312.915	−0.0256456
$531$	0	0
$532$	−4059.28	−0.330812
$533$	31909.8	2.59318
$534$	0	0
$535$	892.932	0.0721585
$536$	2696.49	0.217296
$537$	0	0
$538$	−3806.64	−0.305048
$539$	−4704.58	−0.375957
$540$	0	0
$541$	−14520.4	−1.15394	−0.576968	−	0.816767i	$-0.695764\pi$
$541$	−14520.4	−1.15394	−0.576968	+	0.816767i	$0.695764\pi$
$542$	12971.8	1.02802
$543$	0	0
$544$	−347.142	−0.0273595
$545$	−3618.90	−0.284434
$546$	0	0
$547$	−7693.18	−0.601347	−0.300673	−	0.953727i	$-0.597211\pi$
$547$	−7693.18	−0.601347	−0.300673	+	0.953727i	$0.597211\pi$
$548$	9406.95	0.733294
$549$	0	0
$550$	953.931	0.0739559
$551$	1250.40	0.0966770
$552$	0	0
$553$	−3683.04	−0.283216
$554$	−11695.6	−0.896925
$555$	0	0
$556$	4276.81	0.326218
$557$	3868.38	0.294270	0.147135	−	0.989116i	$-0.452995\pi$
$557$	3868.38	0.294270	0.147135	+	0.989116i	$0.452995\pi$
$558$	0	0
$559$	−29701.6	−2.24730
$560$	785.512	0.0592749
$561$	0	0
$562$	−7472.26	−0.560851
$563$	−5142.68	−0.384970	−0.192485	−	0.981300i	$-0.561655\pi$
$563$	−5142.68	−0.384970	−0.192485	+	0.981300i	$0.561655\pi$
$564$	0	0
$565$	5911.31	0.440160
$566$	−5720.22	−0.424803
$567$	0	0
$568$	2118.13	0.156470
$569$	−12041.7	−0.887198	−0.443599	−	0.896225i	$-0.646299\pi$
$569$	−12041.7	−0.887198	−0.443599	+	0.896225i	$0.646299\pi$
$570$	0	0
$571$	−1372.60	−0.100598	−0.0502990	−	0.998734i	$-0.516017\pi$
$571$	−1372.60	−0.100598	−0.0502990	+	0.998734i	$0.516017\pi$
$572$	−6676.13	−0.488012
$573$	0	0
$574$	7163.07	0.520873
$575$	4904.85	0.355733
$576$	0	0
$577$	−16953.7	−1.22321	−0.611605	−	0.791164i	$-0.709476\pi$
$577$	−16953.7	−1.22321	−0.611605	+	0.791164i	$0.709476\pi$
$578$	−9590.63	−0.690169
$579$	0	0
$580$	−241.966	−0.0173226
$581$	−10576.0	−0.755190
$582$	0	0
$583$	597.000	0.0424103
$584$	7601.80	0.538638
$585$	0	0
$586$	16077.5	1.13337
$587$	−13534.1	−0.951641	−0.475821	−	0.879542i	$-0.657849\pi$
$587$	−13534.1	−0.951641	−0.475821	+	0.879542i	$0.657849\pi$
$588$	0	0
$589$	24271.6	1.69795
$590$	1296.94	0.0904984
$591$	0	0
$592$	4873.21	0.338324
$593$	6652.32	0.460671	0.230336	−	0.973111i	$-0.426018\pi$
$593$	6652.32	0.460671	0.230336	+	0.973111i	$0.426018\pi$
$594$	0	0
$595$	−532.586	−0.0366956
$596$	1247.38	0.0857295
$597$	0	0
$598$	−34326.8	−2.34737
$599$	−19042.5	−1.29892	−0.649461	−	0.760395i	$-0.725006\pi$
$599$	−19042.5	−1.29892	−0.649461	+	0.760395i	$0.725006\pi$
$600$	0	0
$601$	−17074.4	−1.15887	−0.579435	−	0.815018i	$-0.696727\pi$
$601$	−17074.4	−1.15887	−0.579435	+	0.815018i	$0.696727\pi$
$602$	−6667.37	−0.451398
$603$	0	0
$604$	10137.7	0.682943
$605$	4835.03	0.324912
$606$	0	0
$607$	18360.3	1.22771	0.613855	−	0.789419i	$-0.289618\pi$
$607$	18360.3	1.22771	0.613855	+	0.789419i	$0.289618\pi$
$608$	3307.32	0.220608
$609$	0	0
$610$	5389.40	0.357722
$611$	−20744.5	−1.37354
$612$	0	0
$613$	−6500.25	−0.428291	−0.214146	−	0.976802i	$-0.568697\pi$
$613$	−6500.25	−0.428291	−0.214146	+	0.976802i	$0.568697\pi$
$614$	6133.14	0.403116
$615$	0	0
$616$	−1498.65	−0.0980232
$617$	7106.33	0.463679	0.231840	−	0.972754i	$-0.425526\pi$
$617$	7106.33	0.463679	0.231840	+	0.972754i	$0.425526\pi$
$618$	0	0
$619$	19568.0	1.27061	0.635303	−	0.772263i	$-0.280875\pi$
$619$	19568.0	1.27061	0.635303	+	0.772263i	$0.280875\pi$
$620$	−4696.80	−0.304239
$621$	0	0
$622$	19458.4	1.25436
$623$	8931.96	0.574401
$624$	0	0
$625$	625.000	0.0400000
$626$	−1434.25	−0.0915724
$627$	0	0
$628$	−4229.36	−0.268742
$629$	−3304.09	−0.209448
$630$	0	0
$631$	26180.9	1.65174	0.825869	−	0.563862i	$-0.190685\pi$
$631$	26180.9	1.65174	0.825869	+	0.563862i	$0.190685\pi$
$632$	3000.77	0.188868
$633$	0	0
$634$	−6264.10	−0.392396
$635$	1164.79	0.0727924
$636$	0	0
$637$	21572.1	1.34178
$638$	461.637	0.0286464
$639$	0	0
$640$	−640.000	−0.0395285
$641$	24006.3	1.47924	0.739618	−	0.673027i	$-0.235006\pi$
$641$	24006.3	1.47924	0.739618	+	0.673027i	$0.235006\pi$
$642$	0	0
$643$	9311.37	0.571080	0.285540	−	0.958367i	$-0.407827\pi$
$643$	9311.37	0.571080	0.285540	+	0.958367i	$0.407827\pi$
$644$	−7705.64	−0.471498
$645$	0	0
$646$	−2242.40	−0.136573
$647$	−31854.2	−1.93558	−0.967789	−	0.251764i	$-0.918989\pi$
$647$	−31854.2	−1.93558	−0.967789	+	0.251764i	$0.918989\pi$
$648$	0	0
$649$	−2474.38	−0.149658
$650$	−4374.09	−0.263948
$651$	0	0
$652$	10660.6	0.640337
$653$	9712.83	0.582071	0.291035	−	0.956712i	$-0.406000\pi$
$653$	9712.83	0.582071	0.291035	+	0.956712i	$0.406000\pi$
$654$	0	0
$655$	10754.2	0.641531
$656$	−5836.15	−0.347353
$657$	0	0
$658$	−4656.70	−0.275892
$659$	13997.3	0.827399	0.413699	−	0.910414i	$-0.364237\pi$
$659$	13997.3	0.827399	0.413699	+	0.910414i	$0.364237\pi$
$660$	0	0
$661$	27870.0	1.63996	0.819982	−	0.572389i	$-0.193983\pi$
$661$	27870.0	1.63996	0.819982	+	0.572389i	$0.193983\pi$
$662$	−9644.28	−0.566217
$663$	0	0
$664$	8616.83	0.503611
$665$	5074.10	0.295888
$666$	0	0
$667$	2373.61	0.137791
$668$	−6138.02	−0.355520
$669$	0	0
$670$	−3370.61	−0.194356
$671$	−10282.2	−0.591567
$672$	0	0
$673$	12145.8	0.695669	0.347834	−	0.937556i	$-0.386917\pi$
$673$	12145.8	0.695669	0.347834	+	0.937556i	$0.386917\pi$
$674$	−12721.6	−0.727027
$675$	0	0
$676$	21824.3	1.24171
$677$	6579.48	0.373515	0.186758	−	0.982406i	$-0.440202\pi$
$677$	6579.48	0.373515	0.186758	+	0.982406i	$0.440202\pi$
$678$	0	0
$679$	−12938.4	−0.731268
$680$	433.927	0.0244711
$681$	0	0
$682$	8960.86	0.503122
$683$	21314.6	1.19412	0.597058	−	0.802198i	$-0.296336\pi$
$683$	21314.6	1.19412	0.597058	+	0.802198i	$0.296336\pi$
$684$	0	0
$685$	−11758.7	−0.655878
$686$	11578.2	0.644401
$687$	0	0
$688$	5432.28	0.301023
$689$	−2737.44	−0.151362
$690$	0	0
$691$	−14339.9	−0.789458	−0.394729	−	0.918798i	$-0.629161\pi$
$691$	−14339.9	−0.789458	−0.394729	+	0.918798i	$0.629161\pi$
$692$	−8031.38	−0.441195
$693$	0	0
$694$	−20661.3	−1.13010
$695$	−5346.01	−0.291778
$696$	0	0
$697$	3956.98	0.215038
$698$	19261.4	1.04449
$699$	0	0
$700$	−981.890	−0.0530171
$701$	8307.83	0.447621	0.223811	−	0.974633i	$-0.428150\pi$
$701$	8307.83	0.447621	0.223811	+	0.974633i	$0.428150\pi$
$702$	0	0
$703$	31479.0	1.68884
$704$	1221.03	0.0653684
$705$	0	0
$706$	−22790.3	−1.21491
$707$	−6313.37	−0.335840
$708$	0	0
$709$	−7196.53	−0.381201	−0.190600	−	0.981668i	$-0.561043\pi$
$709$	−7196.53	−0.381201	−0.190600	+	0.981668i	$0.561043\pi$
$710$	−2647.67	−0.139951
$711$	0	0
$712$	−7277.37	−0.383049
$713$	46074.2	2.42005
$714$	0	0
$715$	8345.16	0.436491
$716$	−18353.3	−0.957955
$717$	0	0
$718$	3099.28	0.161092
$719$	−4830.88	−0.250572	−0.125286	−	0.992121i	$-0.539985\pi$
$719$	−4830.88	−0.250572	−0.125286	+	0.992121i	$0.539985\pi$
$720$	0	0
$721$	−543.628	−0.0280801
$722$	7646.03	0.394121
$723$	0	0
$724$	63.0711	0.00323759
$725$	302.457	0.0154938
$726$	0	0
$727$	−28019.2	−1.42940	−0.714699	−	0.699432i	$-0.753436\pi$
$727$	−28019.2	−1.42940	−0.714699	+	0.699432i	$0.753436\pi$
$728$	6871.80	0.349843
$729$	0	0
$730$	−9502.25	−0.481773
$731$	−3683.15	−0.186356
$732$	0	0
$733$	−8925.20	−0.449741	−0.224870	−	0.974389i	$-0.572196\pi$
$733$	−8925.20	−0.449741	−0.224870	+	0.974389i	$0.572196\pi$
$734$	−10093.6	−0.507575
$735$	0	0
$736$	6278.21	0.314426
$737$	6430.67	0.321407
$738$	0	0
$739$	−30797.6	−1.53303	−0.766515	−	0.642226i	$-0.778011\pi$
$739$	−30797.6	−1.53303	−0.766515	+	0.642226i	$0.778011\pi$
$740$	−6091.51	−0.302606
$741$	0	0
$742$	−614.497	−0.0304028
$743$	−17118.0	−0.845221	−0.422610	−	0.906312i	$-0.638886\pi$
$743$	−17118.0	−0.845221	−0.422610	+	0.906312i	$0.638886\pi$
$744$	0	0
$745$	−1559.23	−0.0766788
$746$	−20368.1	−0.999635
$747$	0	0
$748$	−827.874	−0.0404680
$749$	1753.52	0.0855438
$750$	0	0
$751$	33246.5	1.61542	0.807711	−	0.589579i	$-0.200706\pi$
$751$	33246.5	1.61542	0.807711	+	0.589579i	$0.200706\pi$
$752$	3794.07	0.183983
$753$	0	0
$754$	−2116.76	−0.102239
$755$	−12672.1	−0.610843
$756$	0	0
$757$	41146.4	1.97555	0.987775	−	0.155888i	$-0.0498239\pi$
$757$	41146.4	1.97555	0.987775	+	0.155888i	$0.0498239\pi$
$758$	−14665.8	−0.702752
$759$	0	0
$760$	−4134.15	−0.197318
$761$	32300.0	1.53860	0.769301	−	0.638887i	$-0.220605\pi$
$761$	32300.0	1.53860	0.769301	+	0.638887i	$0.220605\pi$
$762$	0	0
$763$	−7106.72	−0.337196
$764$	−6390.91	−0.302637
$765$	0	0
$766$	2170.99	0.102404
$767$	11345.8	0.534126
$768$	0	0
$769$	29930.5	1.40354	0.701769	−	0.712404i	$-0.252394\pi$
$769$	29930.5	1.40354	0.701769	+	0.712404i	$0.252394\pi$
$770$	1873.31	0.0876746
$771$	0	0
$772$	2319.47	0.108134
$773$	15385.0	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$773$	15385.0	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$774$	0	0
$775$	5871.00	0.272120
$776$	10541.7	0.487659
$777$	0	0
$778$	855.776	0.0394358
$779$	−37699.3	−1.73391
$780$	0	0
$781$	5051.38	0.231438
$782$	−4256.70	−0.194654
$783$	0	0
$784$	−3945.43	−0.179730
$785$	5286.70	0.240370
$786$	0	0
$787$	−15885.6	−0.719520	−0.359760	−	0.933045i	$-0.617141\pi$
$787$	−15885.6	−0.719520	−0.359760	+	0.933045i	$0.617141\pi$
$788$	8383.85	0.379013
$789$	0	0
$790$	−3750.97	−0.168928
$791$	11608.5	0.521809
$792$	0	0
$793$	47147.5	2.11129
$794$	−5337.69	−0.238574
$795$	0	0
$796$	−6994.31	−0.311440
$797$	33022.3	1.46764	0.733821	−	0.679343i	$-0.237735\pi$
$797$	33022.3	1.46764	0.733821	+	0.679343i	$0.237735\pi$
$798$	0	0
$799$	−2572.42	−0.113900
$800$	800.000	0.0353553
$801$	0	0
$802$	23870.4	1.05099
$803$	18129.0	0.796710
$804$	0	0
$805$	9632.04	0.421720
$806$	−41088.5	−1.79563
$807$	0	0
$808$	5143.85	0.223961
$809$	11391.7	0.495070	0.247535	−	0.968879i	$-0.420379\pi$
$809$	11391.7	0.495070	0.247535	+	0.968879i	$0.420379\pi$
$810$	0	0
$811$	−14491.3	−0.627446	−0.313723	−	0.949515i	$-0.601576\pi$
$811$	−14491.3	−0.627446	−0.313723	+	0.949515i	$0.601576\pi$
$812$	−475.167	−0.0205358
$813$	0	0
$814$	11621.8	0.500421
$815$	−13325.7	−0.572735
$816$	0	0
$817$	35090.4	1.50264
$818$	−1532.85	−0.0655193
$819$	0	0
$820$	7295.19	0.310682
$821$	−16425.1	−0.698221	−0.349110	−	0.937082i	$-0.613516\pi$
$821$	−16425.1	−0.698221	−0.349110	+	0.937082i	$0.613516\pi$
$822$	0	0
$823$	28900.5	1.22407	0.612035	−	0.790831i	$-0.290351\pi$
$823$	28900.5	1.22407	0.612035	+	0.790831i	$0.290351\pi$
$824$	442.924	0.0187257
$825$	0	0
$826$	2546.90	0.107286
$827$	7264.81	0.305468	0.152734	−	0.988267i	$-0.451192\pi$
$827$	7264.81	0.305468	0.152734	+	0.988267i	$0.451192\pi$
$828$	0	0
$829$	−40403.8	−1.69274	−0.846370	−	0.532595i	$-0.821217\pi$
$829$	−40403.8	−1.69274	−0.846370	+	0.532595i	$0.821217\pi$
$830$	−10771.0	−0.450444
$831$	0	0
$832$	−5598.83	−0.233299
$833$	2675.05	0.111266
$834$	0	0
$835$	7672.53	0.317987
$836$	7887.40	0.326305
$837$	0	0
$838$	23268.2	0.959172
$839$	40907.4	1.68329	0.841645	−	0.540032i	$-0.181588\pi$
$839$	40907.4	1.68329	0.841645	+	0.540032i	$0.181588\pi$
$840$	0	0
$841$	−24242.6	−0.993999
$842$	23277.6	0.952732
$843$	0	0
$844$	4490.65	0.183145
$845$	−27280.3	−1.11062
$846$	0	0
$847$	9494.93	0.385183
$848$	500.665	0.0202746
$849$	0	0
$850$	−542.409	−0.0218876
$851$	59755.9	2.40706
$852$	0	0
$853$	−4786.33	−0.192123	−0.0960615	−	0.995375i	$-0.530625\pi$
$853$	−4786.33	−0.192123	−0.0960615	+	0.995375i	$0.530625\pi$
$854$	10583.6	0.424079
$855$	0	0
$856$	−1428.69	−0.0570463
$857$	27781.9	1.10737	0.553683	−	0.832727i	$-0.313222\pi$
$857$	27781.9	1.10737	0.553683	+	0.832727i	$0.313222\pi$
$858$	0	0
$859$	9047.37	0.359362	0.179681	−	0.983725i	$-0.442493\pi$
$859$	9047.37	0.359362	0.179681	+	0.983725i	$0.442493\pi$
$860$	−6790.35	−0.269243
$861$	0	0
$862$	14817.3	0.585473
$863$	−9705.84	−0.382840	−0.191420	−	0.981508i	$-0.561309\pi$
$863$	−9705.84	−0.382840	−0.191420	+	0.981508i	$0.561309\pi$
$864$	0	0
$865$	10039.2	0.394617
$866$	19661.6	0.771512
$867$	0	0
$868$	−9223.49	−0.360675
$869$	7156.33	0.279358
$870$	0	0
$871$	−29486.7	−1.14710
$872$	5790.24	0.224865
$873$	0	0
$874$	40554.8	1.56955
$875$	1227.36	0.0474199
$876$	0	0
$877$	−1350.48	−0.0519981	−0.0259990	−	0.999662i	$-0.508277\pi$
$877$	−1350.48	−0.0519981	−0.0259990	+	0.999662i	$0.508277\pi$
$878$	−7463.36	−0.286875
$879$	0	0
$880$	−1526.29	−0.0584673
$881$	15199.2	0.581243	0.290622	−	0.956838i	$-0.406138\pi$
$881$	15199.2	0.581243	0.290622	+	0.956838i	$0.406138\pi$
$882$	0	0
$883$	−5061.18	−0.192890	−0.0964452	−	0.995338i	$-0.530747\pi$
$883$	−5061.18	−0.192890	−0.0964452	+	0.995338i	$0.530747\pi$
$884$	3796.07	0.144430
$885$	0	0
$886$	16269.8	0.616922
$887$	24143.9	0.913948	0.456974	−	0.889480i	$-0.348933\pi$
$887$	24143.9	0.913948	0.456974	+	0.889480i	$0.348933\pi$
$888$	0	0
$889$	2287.38	0.0862952
$890$	9096.71	0.342609
$891$	0	0
$892$	17810.5	0.668543
$893$	24508.2	0.918406
$894$	0	0
$895$	22941.7	0.856821
$896$	−1256.82	−0.0468609
$897$	0	0
$898$	−35668.5	−1.32547
$899$	2841.16	0.105404
$900$	0	0
$901$	−339.456	−0.0125515
$902$	−13918.2	−0.513776
$903$	0	0
$904$	−9458.09	−0.347977
$905$	−78.8389	−0.00289579
$906$	0	0
$907$	21200.1	0.776116	0.388058	−	0.921635i	$-0.373146\pi$
$907$	21200.1	0.776116	0.388058	+	0.921635i	$0.373146\pi$
$908$	−13086.4	−0.478290
$909$	0	0
$910$	−8589.75	−0.312909
$911$	−31897.6	−1.16006	−0.580030	−	0.814595i	$-0.696959\pi$
$911$	−31897.6	−1.16006	−0.580030	+	0.814595i	$0.696959\pi$
$912$	0	0
$913$	20549.7	0.744901
$914$	−9913.38	−0.358759
$915$	0	0
$916$	309.315	0.0111573
$917$	21119.0	0.760534
$918$	0	0
$919$	11183.2	0.401416	0.200708	−	0.979651i	$-0.435676\pi$
$919$	11183.2	0.401416	0.200708	+	0.979651i	$0.435676\pi$
$920$	−7847.76	−0.281232
$921$	0	0
$922$	−7177.06	−0.256360
$923$	−23162.3	−0.825997
$924$	0	0
$925$	7614.39	0.270659
$926$	21002.1	0.745325
$927$	0	0
$928$	387.145	0.0136947
$929$	−24971.0	−0.881887	−0.440943	−	0.897535i	$-0.645356\pi$
$929$	−24971.0	−0.881887	−0.440943	+	0.897535i	$0.645356\pi$
$930$	0	0
$931$	−25485.9	−0.897173
$932$	16974.4	0.596581
$933$	0	0
$934$	22606.4	0.791973
$935$	1034.84	0.0361957
$936$	0	0
$937$	−26746.4	−0.932514	−0.466257	−	0.884649i	$-0.654398\pi$
$937$	−26746.4	−0.932514	−0.466257	+	0.884649i	$0.654398\pi$
$938$	−6619.14	−0.230408
$939$	0	0
$940$	−4742.58	−0.164560
$941$	−6143.79	−0.212840	−0.106420	−	0.994321i	$-0.533939\pi$
$941$	−6143.79	−0.212840	−0.106420	+	0.994321i	$0.533939\pi$
$942$	0	0
$943$	−71563.6	−2.47130
$944$	−2075.10	−0.0715453
$945$	0	0
$946$	12955.0	0.445248
$947$	−36778.5	−1.26203	−0.631013	−	0.775772i	$-0.717361\pi$
$947$	−36778.5	−1.26203	−0.631013	+	0.775772i	$0.717361\pi$
$948$	0	0
$949$	−83127.4	−2.84344
$950$	5167.69	0.176486
$951$	0	0
$952$	852.138	0.0290104
$953$	−617.414	−0.0209863	−0.0104932	−	0.999945i	$-0.503340\pi$
$953$	−617.414	−0.0209863	−0.0104932	+	0.999945i	$0.503340\pi$
$954$	0	0
$955$	7988.64	0.270687
$956$	−313.978	−0.0106222
$957$	0	0
$958$	−34585.0	−1.16638
$959$	−23091.5	−0.777542
$960$	0	0
$961$	25358.9	0.851227
$962$	−53289.6	−1.78599
$963$	0	0
$964$	−21590.8	−0.721361
$965$	−2899.34	−0.0967181
$966$	0	0
$967$	−1427.65	−0.0474770	−0.0237385	−	0.999718i	$-0.507557\pi$
$967$	−1427.65	−0.0474770	−0.0237385	+	0.999718i	$0.507557\pi$
$968$	−7736.05	−0.256866
$969$	0	0
$970$	−13177.1	−0.436175
$971$	−26134.6	−0.863748	−0.431874	−	0.901934i	$-0.642147\pi$
$971$	−26134.6	−0.863748	−0.431874	+	0.901934i	$0.642147\pi$
$972$	0	0
$973$	−10498.4	−0.345902
$974$	3235.86	0.106451
$975$	0	0
$976$	−8623.04	−0.282804
$977$	−20013.0	−0.655344	−0.327672	−	0.944792i	$-0.606264\pi$
$977$	−20013.0	−0.655344	−0.327672	+	0.944792i	$0.606264\pi$
$978$	0	0
$979$	−17355.3	−0.566575
$980$	4931.79	0.160755
$981$	0	0
$982$	−5483.39	−0.178189
$983$	2113.37	0.0685716	0.0342858	−	0.999412i	$-0.489084\pi$
$983$	2113.37	0.0685716	0.0342858	+	0.999412i	$0.489084\pi$
$984$	0	0
$985$	−10479.8	−0.339000
$986$	−262.489	−0.00847804
$987$	0	0
$988$	−36166.3	−1.16458
$989$	66611.3	2.14167
$990$	0	0
$991$	11029.8	0.353554	0.176777	−	0.984251i	$-0.443433\pi$
$991$	11029.8	0.353554	0.176777	+	0.984251i	$0.443433\pi$
$992$	7514.89	0.240522
$993$	0	0
$994$	−5199.43	−0.165911
$995$	8742.88	0.278561
$996$	0	0
$997$	−10056.0	−0.319435	−0.159717	−	0.987163i	$-0.551058\pi$
$997$	−10056.0	−0.319435	−0.159717	+	0.987163i	$0.551058\pi$
$998$	27514.4	0.872697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.a.u.1.2	yes	4
3.2	odd	2		810.4.a.t.1.2	✓	4
9.2	odd	6		810.4.e.bh.271.3		8
9.4	even	3		810.4.e.bg.541.3		8
9.5	odd	6		810.4.e.bh.541.3		8
9.7	even	3		810.4.e.bg.271.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.4.a.t.1.2	✓	4	3.2	odd	2
810.4.a.u.1.2	yes	4	1.1	even	1	trivial
810.4.e.bg.271.3		8	9.7	even	3
810.4.e.bg.541.3		8	9.4	even	3
810.4.e.bh.271.3		8	9.2	odd	6
810.4.e.bh.541.3		8	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -9.81890 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -9.81890 q^{7} +8.00000 q^{8} -10.0000 q^{10} +19.0786 q^{11} -87.4818 q^{13} -19.6378 q^{14} +16.0000 q^{16} -10.8482 q^{17} +103.354 q^{19} -20.0000 q^{20} +38.1573 q^{22} +196.194 q^{23} +25.0000 q^{25} -174.964 q^{26} -39.2756 q^{28} +12.0983 q^{29} +234.840 q^{31} +32.0000 q^{32} -21.6964 q^{34} +49.0945 q^{35} +304.576 q^{37} +206.708 q^{38} -40.0000 q^{40} -364.759 q^{41} +339.517 q^{43} +76.3145 q^{44} +392.388 q^{46} +237.129 q^{47} -246.589 q^{49} +50.0000 q^{50} -349.927 q^{52} +31.2915 q^{53} -95.3931 q^{55} -78.5512 q^{56} +24.1966 q^{58} -129.694 q^{59} -538.940 q^{61} +469.680 q^{62} +64.0000 q^{64} +437.409 q^{65} +337.061 q^{67} -43.3927 q^{68} +98.1890 q^{70} +264.767 q^{71} +950.225 q^{73} +609.151 q^{74} +413.415 q^{76} -187.331 q^{77} +375.097 q^{79} -80.0000 q^{80} -729.519 q^{82} +1077.10 q^{83} +54.2409 q^{85} +679.035 q^{86} +152.629 q^{88} -909.671 q^{89} +858.975 q^{91} +784.776 q^{92} +474.258 q^{94} -516.769 q^{95} +1317.71 q^{97} -493.179 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8} - 40 q^{10} - 36 q^{11} + 32 q^{13} + 4 q^{14} + 64 q^{16} + 90 q^{17} + 164 q^{19} - 80 q^{20} - 72 q^{22} + 42 q^{23} + 100 q^{25} + 64 q^{26} + 8 q^{28} + 36 q^{29} + 446 q^{31} + 128 q^{32} + 180 q^{34} - 10 q^{35} + 686 q^{37} + 328 q^{38} - 160 q^{40} + 108 q^{41} + 470 q^{43} - 144 q^{44} + 84 q^{46} + 804 q^{47} + 1338 q^{49} + 200 q^{50} + 128 q^{52} + 168 q^{53} + 180 q^{55} + 16 q^{56} + 72 q^{58} + 768 q^{59} + 596 q^{61} + 892 q^{62} + 256 q^{64} - 160 q^{65} - 424 q^{67} + 360 q^{68} - 20 q^{70} + 348 q^{71} - 94 q^{73} + 1372 q^{74} + 656 q^{76} + 630 q^{77} + 1088 q^{79} - 320 q^{80} + 216 q^{82} + 522 q^{83} - 450 q^{85} + 940 q^{86} - 288 q^{88} + 1128 q^{89} + 2842 q^{91} + 168 q^{92} + 1608 q^{94} - 820 q^{95} + 2486 q^{97} + 2676 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8 - 40 * q^10 - 36 * q^11 + 32 * q^13 + 4 * q^14 + 64 * q^16 + 90 * q^17 + 164 * q^19 - 80 * q^20 - 72 * q^22 + 42 * q^23 + 100 * q^25 + 64 * q^26 + 8 * q^28 + 36 * q^29 + 446 * q^31 + 128 * q^32 + 180 * q^34 - 10 * q^35 + 686 * q^37 + 328 * q^38 - 160 * q^40 + 108 * q^41 + 470 * q^43 - 144 * q^44 + 84 * q^46 + 804 * q^47 + 1338 * q^49 + 200 * q^50 + 128 * q^52 + 168 * q^53 + 180 * q^55 + 16 * q^56 + 72 * q^58 + 768 * q^59 + 596 * q^61 + 892 * q^62 + 256 * q^64 - 160 * q^65 - 424 * q^67 + 360 * q^68 - 20 * q^70 + 348 * q^71 - 94 * q^73 + 1372 * q^74 + 656 * q^76 + 630 * q^77 + 1088 * q^79 - 320 * q^80 + 216 * q^82 + 522 * q^83 - 450 * q^85 + 940 * q^86 - 288 * q^88 + 1128 * q^89 + 2842 * q^91 + 168 * q^92 + 1608 * q^94 - 820 * q^95 + 2486 * q^97 + 2676 * q^98

Introduction

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